ABSTRACT: A sketch of the origins of mathematics that
explains what is distinctive about mathematical knowledge
but does not essentially depend on a reference to
traditional mathematicals (such as numbers). This
explanation explains why mathematical experience
supports Platonism but also why Platonism can likely be
never more than a regulative idea(l).
Feferman's questions about the distinctiveness of
mathematical thought are put to Hersh/Lakatos, and
answered via Rota's phenomenological thinking, and
answered in a way without undercutting the phenomena
which fund Platonism, but which at the same time reveal
the extreme difficulties in sustaining Platonism as anything
more than a regulative idea(l). [End of Abstract.]
I've been enjoying Hersh's _What is Mathematics,
Really?_ I think that Reuben Hersh is importantly
correcting the course of the philosophy of mathematics,
but at the same time he _over-corrects_ it.
How should the over-correction be itself corrected?
Hersh and Lakatos stand in need of such correction,
and in need of the very same correction -- they both overly
collapse -- in a phenomenologically undiscerning way --
uses of 'certainty' (and its "cognates").
I've been writing a long review essay on the
phenomenological studies of mathematical thought in
Gian-Carlo Rota's _Indiscrete Thoughts_. It is worth
observing that, although Rota and Hersh, through
forewards and dust-jacket copy, strongly applaud one
another's (philosophical) work, Rota's phenomenological
appreciation of mathematical proof and mathematical
understanding provides the needed correction to Hersh's
course.
I want to try to explain this here, and do so in a way
that points to some answers to questions raised by Sol
Feferman.
In a recent FOM posting, Cook asked Hersh how
mathematics is distinguished from other "academic
subjects". Hersh answered that mathematics was one of
the "humanities" (because it is entirely a human product)
and is distinguished among the humanities by being about
mathematicals.
In his review of Lakatos, Sol Feferman asked a series
of questions -- one close to that of Cook's -- which we,
qua phenomenological Rota, put to Hersh (but don't wait
around for his answer), --
FEFERMAN'S QUESTIONS:
[1] What is distinctive about mathematics?
_i.e._,
[2] What is distinctive about its verification structure?
[3] What is distinctive about its conceptual content?
The absolutely important _necessary condition_ for giving
a good answer to these questions:
Necessary Condition: They must allow us to
characterize mathematics in a way that is independent of
the kind of appeal Hersh makes, viz., that mathematics
studies the mathematicals (studies what mathematicians
study).
It is exactly by evading this issue that Hersh misses
the distinctive features of the verification structure of
mathematics (or: is able to play a bit too much of the old
fast and loose with it).
****
I'll make this short, though details can be supplied (some
will be supplied in a posting giving an account of
Lebesgue's conception of arithmetic).
First, I'll describe the two stages in the origin of
mathematics, and then I'll give an example.
[1] First stage of the origin of mathematics: Witty persons
become aware of (practical) problems which have
solutions which (a) can be framed or represented in
thought and (b) can be seen by thought alone to be
definitive solutions (seen by a peculiar sort of light;
mathematical proofs will be architectures in that light).
The recognition and cultivation of such problems is the
first stage in the development of mathematical thought.
Their cultivation may go beyond practical interests (e.g.,
as in play or poetry. . .riddles. . .problems to be solved in
contests, etc. . . .N.B., Ian Hacking's phantasy of
language issuing more from play rather than work).
[2] Second stage of the origin of mathematics (Berkeleyian
abstraction): it is observed that the light by which one sees
that such and such is THE solution to the problem "Q?"
has an authority that is not bound to the concrete
particulars of the problem. In successfully reframing such
problems, their solutions, and the
exhibition/demonstration of such solutions as _the_
solutions, mathematics proper begins.
EXAMPLE: [This example is meant to illustrate the ideas,
so it is ideal; but it is exemplary, too, in that more
realistic examples having to do with the "real" origins of
arithmetic, algebra, geometry, combinatorics. . . will be
patterned after it.]
[E1] It becomes practically important for me to know the
minimum number of fruitfly I have to capture in order to
be certain that I have two which are the same sex.
I proceed experimentally.
First, I form collections each containing one fruitfly.
I notce that none of them contains two fruitfly which have
the same sex (but dull empiricist that I am, I do not notice
that each collection fails to contain two fruitfly of the same
sex _because_ each collection contains only one fruitfly).
Second, I form collections each containing _three_
fruitfly (because dull empiricist that I am I don't see that
the next logical step would be to try collections of two). I
find that each collection contains at least two fruitfly of the
same sex. But now I worry whether this is the least
number. And dull empiricst that I am, I first make a lot
of collections containing _two_ fruitfly and a lot of
collections containing _five_ fruitfly, comparing them to
the collections containing three fruitfly to see whether the
collections containing five fruitfly or the collections
containing two fruitfly contain fewer fruitfly than the
collections containing three fruitfly. I learn that the
collections containing two fruitfly contain fewer fruitfly;
so I will not look through those collections to see if two
fruitfly of the same sex invariably occur. I find that they
do not. So I conclude: _3_ is the answer to the initial
question. Of course, it might have happened that male
fruitfly were very rare, so that in fact all the pairs I
collected were pairs of females, and I was led to answer:
2, instead of 3. But in any case, someone will cast
doubt on my answer _3_ because I overlooked so many
other possibilities, such as 4, 23, 197, 272. . .
Observe that it is rather unlikely that there should be
such very dull empiricists. . .who never let in any as it
were _apriori_ thinking, for whom it could never be
decisive that collections containing two things have fewer
things than those containing three or four things.
But here is the example of _apriori_ thinking I want
to dwell on:
A collection of three firefly must contain at least two
which have the same sex, for here are the only possible
combinations of three firefly identified up to sex (M, F):
MMM, MMF, MFF, FFF.
"Check that this lists all possible combinations."
(There are of course a number of ways that the check
can be made _apriori_ . . .)
Here one has solved the problem, definitively, and
_apriori_. Any skeptic must be either witless or fail to
understand the terms of the problem (such as its being
assumed that all firefly are either M or F -- see below ON
HIDDEN ASSUMPTIONS).
This problem illustrates the first stage. The second
stage, mathematics proper, begins with the observation
that in dmeonstrating to onesself that the solution to the
problem is three, one actually has proved a more general
proposition (which in the framing becomes more abstract):
The demonstration of the correctness of the solution
to the problem does not essentially depend on fruitflies or
on sex.
Ginning up concepts to state and prove, and then
stating and proving, the more general/abstract proposition
is the very essence of logico-mathematical activity. It is
one thing to notice that the "proof" proves more than the
particular proposition (about collections of sexed fruitfly)
at issue. That's the first phase of the second stage on the
way to mathematics. The second phase is to find the
concepts in which to frame the more general/abstract
proposition. For examples, a language containing
'individual', 'property', 'collection', 'has the property',
'does not have the property', and so on, and more or less
explicit rules for using these terms.
Some conjectures about the characteristic trait of
mathematical concepts:
What enable emergence of the definitive solution to
the firefly problem?
Being able to canvas all the possibilities in advance.
This suggests then a tentative characterization of
mathematical concepts: that they enable us to canvass
(directly as above, or indirectly, as is more typical) in
advance all the relevant possibilities of what they frame (of
the problems which can be framed through them).
This suggests why Platonism might seem supported
but in actuality false:
Concepts arising in mathematics (through the process
of Berkleyian "abstraction/generalizatiuon" sketched
above) may sustain _apriori_ solutions to a wide range of
problems posed through them (the concepts), but the
concepts may also be rough around the edges -- we in fact
cannot canvass all the relevant possibilities relating to all
problems framed in those concepts because the concepts
are not decisive on all such ranges.
REMARKS:
Rota calls Evidenz (which he translates "insightful;
understanding") the kind of demonstration/light by which
we decisively (and _apriori_) find proposed solutions to
mathematical problems to be solutions to problems. Yes,
one can always be "skeptical" in the sense of demanding
greater explicitness, etc. But there comes a point at
which one either has understood the problem or one has
not. . .whereafter we have to say that, as far as the
problem at issue is concerned, the skeptic is not cooking
on all four burners. (For those who can read Descartes'
First meditation with great care, it can be seen that
Descartes -- like Wittgenstein and Cavell -- makes
essentially this point. . .that doubt uninhibited by authentic
understanding is madness, skeptical terror only.)
I had best break off here.
rbrt tragesser